on perfectness of dot product graph of a commutative ring

Let a be a commutative ring with nonzero identity, and 1 ≤ n < ∞ be an integer, and r = a × a × · · · × a (n times). the total dot product graph of r is the (undirected) graph td(r) with vertices r∗ = r {(0, 0, . . . , 0)}, and two distinct vertices x and y are adjacent if and only if x · y = 0 ∈ a (where x · y denote the normal dot product of x and y). let z(r) denote the set of all zero-divisors of r. then the zero-divisor dot product graph of r is the induced subgraph zd(r) of td(r) with vertices z(r)∗ = z(r) {(0, 0, . . . , 0)}. it follows that if γ(a) is not perfect, then zd(r) (and hence td(r)) is not perfect. in this paper we investigate perfectness of the graphs td(r) and zd(r).